Position and Momentum in Quantum Theory

“In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum mechanics is a different fundamental formalism, in which observables such as position and momentum are not real numbers but operators; consequently there are uncertainty relations, e.g. Δx Δp ~ h, which say that as some observables become more sharply defined, others become more uncertain. Experiments show that quantum mechanics, not classical mechanics, is the correct description of nature.”

“In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object. For example, a heavy truck moving rapidly has a large momentum – it takes a large or prolonged force to get the truck up to this speed, and it takes a large or prolonged force to bring it to a stop afterwards. If the truck were lighter or moving more slowly, then it would have less momentum. […] In quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberguncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.”

“The first use of „momentum“ in its proper mathematical sense is not clear but by the time of Jenning’s Miscellanea in 1721, four years before the final edition of Newton’s Principia Mathematica, momentum M or „quantity of motion“ was being defined for students as „a rectangle“, the product of Q and V, where Q is „quantity of material“ and V is „velocity“, s/t.”